Science:Math Exam Resources/Courses/MATH100/December 2018/Question 05 (c)/Solution 1

We have $f''(x)={\frac {9}{7}}\cdot {\frac {1\cdot x^{\frac {5}{7}}-(x+1)\cdot {\frac {5}{7x^{\frac {2}{7}}}}}{x^{\frac {10}{7}}}}={\frac {9}{7}}\cdot {\frac {7x-5(x+1)}{7x^{\frac {12}{7}}}}={\frac {9(2x-5)}{49x^{\frac {12}{7}}}}.$ This denominator will always be nonnegative (because of the even number $12$ in the exponent), so it suffices to understand when the numerator is positive/negative. Note also that $f''(x)$ is undefined at $x=0$. The numerator is zero when $x=5/2$, positive when $x>5/2$, and negative when $x<5/2$. We therefore conclude that $\color {blue}f(x){\text{ is concave up on }}\left({\frac {5}{2}},+\infty \right){\text{ and concave down on }}\left(-\infty ,0\right)\cup \left(0,{\frac {5}{2}}\right)$.